Question

Find the linear function that is the best fit for the data.
x
1
3
5
7
9

y
2
3
4
5
6
Question content area bottom
Part 1
What is the linear function of the​ data?
yequals
  
enter your response herexplus​(
  
enter your response here​)
​(Type integers or​ decimals.)

Answer 1

To find the linear function that best fits the given data points, we can use the method of linear regression. The data points provided are:

\[
\begin{align*}
(1, 2) \\
(3, 3) \\
(5, 4) \\
(7, 5) \\
(9, 6) \\
\end{align*}
\]

We want to find the equation of the line in the form:

\[
y = mx + b
\]

where \( m \) is the slope and \( b \) is the y-intercept.

### Step 1: Calculate the slope \( m \)

The formula for the slope \( m \) is:

\[
m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2}
\]

Where:
- \( N \) is the number of data points
- \( x \) is the independent variable and \( y \) is the dependent variable

Let's calculate the necessary components:

\[
\begin{align*}
\sum x & = 1 + 3 + 5 + 7 + 9 = 25 \\
\sum y & = 2 + 3 + 4 + 5 + 6 = 20 \\
\sum xy & = (1 \cdot 2) + (3 \cdot 3) + (5 \cdot 4) + (7 \cdot 5) + (9 \cdot 6) = 2 + 9 + 20 + 35 + 54 = 120 \\
\sum x^2 & = 1^2 + 3^2 + 5^2 + 7^2 + 9^2 = 1 + 9 + 25 + 49 + 81 = 165 \\
\end{align*}
\]

Now, plug everything into the slope formula:

\[
m = \frac{5(120) - (25)(20)}{5(165) - (25)^2} = \frac{600 - 500}{825 - 625} = \frac{100}{200} = 0.5
\]

### Step 2: Calculate the y-intercept \( b \)

The formula for the y-intercept \( b \) is:

\[
b = \frac{\sum y - m(\sum x)}{N}
\]

Plugging in the values we calculated:

\[
b = \frac{20 - 0.5(25)}{5} = \frac{20 - 12.5}{5} = \frac{7.5}{5} = 1.5
\]

### Step 3: Write the linear function

Now, we can formulate the linear function:

\[
y = 0.5x + 1.5
\]

### Final Answer

Thus, the linear function that best fits the data is:

\[
y = 0.5x + 1.5
\]